Remark A.3 This result is nothing more than Corollary 2.2 of [1]}, which is based on their key result Theorem 2.1. We refer to their paper for the proof. But let's remark on what small proof modifications we need to adapt their result to our Theorem REF . The major difference between our stated result and their Corollary 2.2 is the condition on the constant \(A\)  in the covering number bound. Their Corollary 2.2 requires \(A \ge e\)  but we do not impose this requirement here. Indeed, from Lemma REF (i), it is unnatural to require that \(A_n \ge e\)  for all \(n\) , especially since we only have an upper bound for \(V(\mathcal {G}_n)\)  and not a lower bound. For their proofs, the authors only require the condition \(A \ge e\)  to arrive at the uniform entropy integral condition \(J(\delta , \mathcal {F}) \lesssim \delta \sqrt{\nu \log (A / \delta )}\) . From Lemma REF (iv), we have instead a slightly larger bound of \(J(\delta , \mathcal {G}_n) \le \delta \sqrt{\nu _n} ( 1 + \sqrt{1 + \log (A_n / \delta )} )\) . Consequently and by inspecting the proofs of their Corollary 2.2, it suffices to replace their definition of \(K_n = c \nu (\log n \vee \log \frac{A b}{\sigma } )\)  with our slightly larger \(K_n\) , then the remainder of their proof goes through to our case.

